LMFDB - Embedded newform 4032.2.b.c.3583.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4032,2,Mod(3583,4032)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4032, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4032.3583");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4032 = 2^{6} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4032.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$32.1956820950$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 336)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3583.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	4032.3583
Dual form			4032.2.b.c.3583.2

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.00000 - 1.73205i) q^{7} +O(q^{10})$	$q+(-2.00000 - 1.73205i) q^{7} -3.46410i q^{11} +6.92820i q^{17} +4.00000 q^{19} +3.46410i q^{23} +5.00000 q^{25} -6.00000 q^{29} +4.00000 q^{31} +2.00000 q^{37} +6.92820i q^{41} -3.46410i q^{43} +(1.00000 + 6.92820i) q^{49} -6.00000 q^{53} +12.0000 q^{59} -13.8564i q^{61} -3.46410i q^{67} -10.3923i q^{71} -13.8564i q^{73} +(-6.00000 + 6.92820i) q^{77} +10.3923i q^{79} +12.0000 q^{83} -6.92820i q^{89} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{7}+O(q^{10}) $ 2 * q - 4 * q^7	$ 2 q - 4 q^{7} + 8 q^{19} + 10 q^{25} - 12 q^{29} + 8 q^{31} + 4 q^{37} + 2 q^{49} - 12 q^{53} + 24 q^{59} - 12 q^{77} + 24 q^{83}+O(q^{100}) $ 2 * q - 4 * q^7 + 8 * q^19 + 10 * q^25 - 12 * q^29 + 8 * q^31 + 4 * q^37 + 2 * q^49 - 12 * q^53 + 24 * q^59 - 12 * q^77 + 24 * q^83

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4032\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$1793$	$3781$
$\chi(n)$	$-1$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$		0			0		1.00000			$0$
$5$		0			0		−1.00000			$\pi$
$6$		0			0
$7$	−2.00000	−	1.73205i	−0.755929	−	0.654654i
$7$	−2.00000	−	1.73205i	−0.755929	−	0.654654i
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	3.46410i		−	1.04447i	−0.852803	−	0.522233i	$-0.825099\pi$
$11$		−	3.46410i		−	1.04447i	0.852803	−	0.522233i	$-0.174901\pi$
$12$		0			0
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			6.92820i			1.68034i	0.542326	+	0.840168i	$0.317544\pi$
$17$			6.92820i			1.68034i	−0.542326	+	0.840168i	$0.682456\pi$
$18$		0			0
$19$	4.00000			0.917663			0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000			0.917663			0.458831	+	0.888523i	$0.348268\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			3.46410i			0.722315i	0.932505	+	0.361158i	$0.117618\pi$
$23$			3.46410i			0.722315i	−0.932505	+	0.361158i	$0.882382\pi$
$24$		0			0
$25$	5.00000			1.00000
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−6.00000			−1.11417			−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000			−1.11417			−0.557086	+	0.830455i	$0.688081\pi$
$30$		0			0
$31$	4.00000			0.718421			0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000			0.718421			0.359211	+	0.933257i	$0.383046\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	2.00000			0.328798			0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000			0.328798			0.164399	+	0.986394i	$0.447432\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			6.92820i			1.08200i	0.841021	+	0.541002i	$0.181955\pi$
$41$			6.92820i			1.08200i	−0.841021	+	0.541002i	$0.818045\pi$
$42$		0			0
$43$		−	3.46410i		−	0.528271i	−0.964486	−	0.264135i	$-0.914913\pi$
$43$		−	3.46410i		−	0.528271i	0.964486	−	0.264135i	$-0.0850865\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$		0			0
$49$	1.00000	+	6.92820i	0.142857	+	0.989743i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−6.00000			−0.824163			−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000			−0.824163			−0.412082	+	0.911147i	$0.635198\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	12.0000			1.56227			0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000			1.56227			0.781133	+	0.624364i	$0.214642\pi$
$60$		0			0
$61$		−	13.8564i		−	1.77413i	−0.461644	−	0.887066i	$-0.652740\pi$
$61$		−	13.8564i		−	1.77413i	0.461644	−	0.887066i	$-0.347260\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$		−	3.46410i		−	0.423207i	−0.977356	−	0.211604i	$-0.932131\pi$
$67$		−	3.46410i		−	0.423207i	0.977356	−	0.211604i	$-0.0678686\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	10.3923i		−	1.23334i	−0.787222	−	0.616670i	$-0.788481\pi$
$71$		−	10.3923i		−	1.23334i	0.787222	−	0.616670i	$-0.211519\pi$
$72$		0			0
$73$		−	13.8564i		−	1.62177i	−0.585206	−	0.810885i	$-0.698986\pi$
$73$		−	13.8564i		−	1.62177i	0.585206	−	0.810885i	$-0.301014\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−6.00000	+	6.92820i	−0.683763	+	0.789542i
$78$		0			0
$79$			10.3923i			1.16923i	0.811312	+	0.584613i	$0.198754\pi$
$79$			10.3923i			1.16923i	−0.811312	+	0.584613i	$0.801246\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	12.0000			1.31717			0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000			1.31717			0.658586	+	0.752506i	$0.271155\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	6.92820i		−	0.734388i	−0.930144	−	0.367194i	$-0.880318\pi$
$89$		−	6.92820i		−	0.734388i	0.930144	−	0.367194i	$-0.119682\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	13.8564i		−	1.37876i	−0.724398	−	0.689382i	$-0.757882\pi$
$101$		−	13.8564i		−	1.37876i	0.724398	−	0.689382i	$-0.242118\pi$
$102$		0			0
$103$	4.00000			0.394132			0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000			0.394132			0.197066	+	0.980390i	$0.436859\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	17.3205i		−	1.67444i	−0.546869	−	0.837218i	$-0.684180\pi$
$107$		−	17.3205i		−	1.67444i	0.546869	−	0.837218i	$-0.315820\pi$
$108$		0			0
$109$	10.0000			0.957826			0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000			0.957826			0.478913	+	0.877862i	$0.341031\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	6.00000			0.564433			0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000			0.564433			0.282216	+	0.959351i	$0.408930\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	12.0000	−	13.8564i	1.10004	−	1.27021i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	17.3205i		−	1.53695i	−0.639882	−	0.768473i	$-0.721017\pi$
$127$		−	17.3205i		−	1.53695i	0.639882	−	0.768473i	$-0.278983\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−12.0000			−1.04844			−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000			−1.04844			−0.524222	+	0.851581i	$0.675644\pi$
$132$		0			0
$133$	−8.00000	−	6.92820i	−0.693688	−	0.600751i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−18.0000			−1.53784			−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000			−1.53784			−0.768922	+	0.639343i	$0.779207\pi$
$138$		0			0
$139$	−4.00000			−0.339276			−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000			−0.339276			−0.169638	+	0.985506i	$0.554260\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	18.0000			1.47462			0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000			1.47462			0.737309	+	0.675556i	$0.236096\pi$
$150$		0			0
$151$		−	3.46410i		−	0.281905i	−0.990016	−	0.140952i	$-0.954984\pi$
$151$		−	3.46410i		−	0.281905i	0.990016	−	0.140952i	$-0.0450164\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$			13.8564i			1.10586i	0.833227	+	0.552931i	$0.186491\pi$
$157$			13.8564i			1.10586i	−0.833227	+	0.552931i	$0.813509\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	6.00000	−	6.92820i	0.472866	−	0.546019i
$162$		0			0
$163$		−	3.46410i		−	0.271329i	−0.990755	−	0.135665i	$-0.956683\pi$
$163$		−	3.46410i		−	0.271329i	0.990755	−	0.135665i	$-0.0433170\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	24.0000			1.85718			0.928588	−	0.371113i	$-0.121024\pi$
$167$	24.0000			1.85718			0.928588	+	0.371113i	$0.121024\pi$
$168$		0			0
$169$	13.0000			1.00000
$170$		0			0
$171$		0			0
$172$		0			0
$173$			13.8564i			1.05348i	0.850026	+	0.526742i	$0.176586\pi$
$173$			13.8564i			1.05348i	−0.850026	+	0.526742i	$0.823414\pi$
$174$		0			0
$175$	−10.0000	−	8.66025i	−0.755929	−	0.654654i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	3.46410i		−	0.258919i	−0.991585	−	0.129460i	$-0.958676\pi$
$179$		−	3.46410i		−	0.258919i	0.991585	−	0.129460i	$-0.0413242\pi$
$180$		0			0
$181$			13.8564i			1.02994i	0.857209	+	0.514969i	$0.172197\pi$
$181$			13.8564i			1.02994i	−0.857209	+	0.514969i	$0.827803\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	24.0000			1.75505
$188$		0			0
$189$		0			0
$190$		0			0
$191$			17.3205i			1.25327i	0.779314	+	0.626634i	$0.215568\pi$
$191$			17.3205i			1.25327i	−0.779314	+	0.626634i	$0.784432\pi$
$192$		0			0
$193$	−14.0000			−1.00774			−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000			−1.00774			−0.503871	+	0.863779i	$0.668091\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	18.0000			1.28245			0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000			1.28245			0.641223	+	0.767354i	$0.278427\pi$
$198$		0			0
$199$	20.0000			1.41776			0.708881	−	0.705328i	$-0.249200\pi$
$199$	20.0000			1.41776			0.708881	+	0.705328i	$0.249200\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	12.0000	+	10.3923i	0.842235	+	0.729397i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	13.8564i		−	0.958468i
$210$		0			0
$211$			10.3923i			0.715436i	0.933830	+	0.357718i	$0.116445\pi$
$211$			10.3923i			0.715436i	−0.933830	+	0.357718i	$0.883555\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−8.00000	−	6.92820i	−0.543075	−	0.470317i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−4.00000			−0.267860			−0.133930	−	0.990991i	$-0.542760\pi$
$223$	−4.00000			−0.267860			−0.133930	+	0.990991i	$0.542760\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−12.0000			−0.796468			−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000			−0.796468			−0.398234	+	0.917284i	$0.630377\pi$
$228$		0			0
$229$			13.8564i			0.915657i	0.889041	+	0.457829i	$0.151373\pi$
$229$			13.8564i			0.915657i	−0.889041	+	0.457829i	$0.848627\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−18.0000			−1.17922			−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000			−1.17922			−0.589610	+	0.807688i	$0.700718\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	24.2487i		−	1.56852i	−0.620433	−	0.784259i	$-0.713043\pi$
$239$		−	24.2487i		−	1.56852i	0.620433	−	0.784259i	$-0.286957\pi$
$240$		0			0
$241$			13.8564i			0.892570i	0.894891	+	0.446285i	$0.147253\pi$
$241$			13.8564i			0.892570i	−0.894891	+	0.446285i	$0.852747\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	12.0000			0.757433			0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000			0.757433			0.378717	+	0.925513i	$0.376365\pi$
$252$		0			0
$253$	12.0000			0.754434
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	6.92820i		−	0.432169i	−0.976375	−	0.216085i	$-0.930671\pi$
$257$		−	6.92820i		−	0.432169i	0.976375	−	0.216085i	$-0.0693287\pi$
$258$		0			0
$259$	−4.00000	−	3.46410i	−0.248548	−	0.215249i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	10.3923i		−	0.640817i	−0.947279	−	0.320408i	$-0.896180\pi$
$263$		−	10.3923i		−	0.640817i	0.947279	−	0.320408i	$-0.103820\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$			13.8564i			0.844840i	0.906400	+	0.422420i	$0.138819\pi$
$269$			13.8564i			0.844840i	−0.906400	+	0.422420i	$0.861181\pi$
$270$		0			0
$271$	28.0000			1.70088			0.850439	−	0.526073i	$-0.176336\pi$
$271$	28.0000			1.70088			0.850439	+	0.526073i	$0.176336\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	17.3205i		−	1.04447i
$276$		0			0
$277$	−22.0000			−1.32185			−0.660926	−	0.750451i	$-0.729836\pi$
$277$	−22.0000			−1.32185			−0.660926	+	0.750451i	$0.729836\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	30.0000			1.78965			0.894825	−	0.446417i	$-0.147300\pi$
$281$	30.0000			1.78965			0.894825	+	0.446417i	$0.147300\pi$
$282$		0			0
$283$	4.00000			0.237775			0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000			0.237775			0.118888	+	0.992908i	$0.462067\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	12.0000	−	13.8564i	0.708338	−	0.817918i
$288$		0			0
$289$	−31.0000			−1.82353
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−6.00000	+	6.92820i	−0.345834	+	0.399335i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−20.0000			−1.14146			−0.570730	−	0.821138i	$-0.693340\pi$
$307$	−20.0000			−1.14146			−0.570730	+	0.821138i	$0.693340\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$			13.8564i			0.783210i	0.920133	+	0.391605i	$0.128080\pi$
$313$			13.8564i			0.783210i	−0.920133	+	0.391605i	$0.871920\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	18.0000			1.01098			0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000			1.01098			0.505490	+	0.862832i	$0.331312\pi$
$318$		0			0
$319$			20.7846i			1.16371i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			27.7128i			1.54198i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		−	31.1769i		−	1.71364i	−0.515617	−	0.856819i	$-0.672437\pi$
$331$		−	31.1769i		−	1.71364i	0.515617	−	0.856819i	$-0.327563\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	2.00000			0.108947			0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000			0.108947			0.0544735	+	0.998515i	$0.482652\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	13.8564i		−	0.750366i
$342$		0			0
$343$	10.0000	−	15.5885i	0.539949	−	0.841698i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	17.3205i		−	0.929814i	−0.885360	−	0.464907i	$-0.846088\pi$
$347$		−	17.3205i		−	0.929814i	0.885360	−	0.464907i	$-0.153912\pi$
$348$		0			0
$349$		−	27.7128i		−	1.48343i	−0.670714	−	0.741716i	$-0.734012\pi$
$349$		−	27.7128i		−	1.48343i	0.670714	−	0.741716i	$-0.265988\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	20.7846i		−	1.10625i	−0.833097	−	0.553127i	$-0.813435\pi$
$353$		−	20.7846i		−	1.10625i	0.833097	−	0.553127i	$-0.186565\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$			3.46410i			0.182828i	0.995813	+	0.0914141i	$0.0291387\pi$
$359$			3.46410i			0.182828i	−0.995813	+	0.0914141i	$0.970861\pi$
$360$		0			0
$361$	−3.00000			−0.157895
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−28.0000			−1.46159			−0.730794	−	0.682598i	$-0.760850\pi$
$367$	−28.0000			−1.46159			−0.730794	+	0.682598i	$0.760850\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	12.0000	+	10.3923i	0.623009	+	0.539542i
$372$		0			0
$373$	10.0000			0.517780			0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000			0.517780			0.258890	+	0.965907i	$0.416643\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		−	3.46410i		−	0.177939i	−0.996034	−	0.0889695i	$-0.971643\pi$
$379$		−	3.46410i		−	0.177939i	0.996034	−	0.0889695i	$-0.0283574\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−24.0000			−1.22634			−0.613171	−	0.789950i	$-0.710106\pi$
$383$	−24.0000			−1.22634			−0.613171	+	0.789950i	$0.710106\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−6.00000			−0.304212			−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000			−0.304212			−0.152106	+	0.988364i	$0.548606\pi$
$390$		0			0
$391$	−24.0000			−1.21373
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$			27.7128i			1.39087i	0.718591	+	0.695433i	$0.244787\pi$
$397$			27.7128i			1.39087i	−0.718591	+	0.695433i	$0.755213\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	6.00000			0.299626			0.149813	−	0.988714i	$-0.452133\pi$
$401$	6.00000			0.299626			0.149813	+	0.988714i	$0.452133\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	6.92820i		−	0.343418i
$408$		0			0
$409$			13.8564i			0.685155i	0.939490	+	0.342578i	$0.111300\pi$
$409$			13.8564i			0.685155i	−0.939490	+	0.342578i	$0.888700\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−24.0000	−	20.7846i	−1.18096	−	1.02274i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	12.0000			0.586238			0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000			0.586238			0.293119	+	0.956076i	$0.405307\pi$
$420$		0			0
$421$	−22.0000			−1.07221			−0.536107	−	0.844150i	$-0.680106\pi$
$421$	−22.0000			−1.07221			−0.536107	+	0.844150i	$0.680106\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			34.6410i			1.68034i
$426$		0			0
$427$	−24.0000	+	27.7128i	−1.16144	+	1.34112i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			31.1769i			1.50174i	0.660451	+	0.750870i	$0.270365\pi$
$431$			31.1769i			1.50174i	−0.660451	+	0.750870i	$0.729635\pi$
$432$		0			0
$433$		−	13.8564i		−	0.665896i	−0.942945	−	0.332948i	$-0.891957\pi$
$433$		−	13.8564i		−	0.665896i	0.942945	−	0.332948i	$-0.108043\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			13.8564i			0.662842i
$438$		0			0
$439$	20.0000			0.954548			0.477274	−	0.878755i	$-0.341625\pi$
$439$	20.0000			0.954548			0.477274	+	0.878755i	$0.341625\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	31.1769i		−	1.48126i	−0.671913	−	0.740630i	$-0.734527\pi$
$443$		−	31.1769i		−	1.48126i	0.671913	−	0.740630i	$-0.265473\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	30.0000			1.41579			0.707894	−	0.706319i	$-0.249646\pi$
$449$	30.0000			1.41579			0.707894	+	0.706319i	$0.249646\pi$
$450$		0			0
$451$	24.0000			1.13012
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	26.0000			1.21623			0.608114	−	0.793849i	$-0.291926\pi$
$457$	26.0000			1.21623			0.608114	+	0.793849i	$0.291926\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	13.8564i		−	0.645357i	−0.946509	−	0.322679i	$-0.895417\pi$
$461$		−	13.8564i		−	0.645357i	0.946509	−	0.322679i	$-0.104583\pi$
$462$		0			0
$463$			10.3923i			0.482971i	0.970404	+	0.241486i	$0.0776347\pi$
$463$			10.3923i			0.482971i	−0.970404	+	0.241486i	$0.922365\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−12.0000			−0.555294			−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000			−0.555294			−0.277647	+	0.960683i	$0.589555\pi$
$468$		0			0
$469$	−6.00000	+	6.92820i	−0.277054	+	0.319915i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−12.0000			−0.551761
$474$		0			0
$475$	20.0000			0.917663
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$		−	31.1769i		−	1.41276i	−0.707832	−	0.706380i	$-0.750327\pi$
$487$		−	31.1769i		−	1.41276i	0.707832	−	0.706380i	$-0.249673\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	3.46410i		−	0.156333i	−0.996940	−	0.0781664i	$-0.975093\pi$
$491$		−	3.46410i		−	0.156333i	0.996940	−	0.0781664i	$-0.0249065\pi$
$492$		0			0
$493$		−	41.5692i		−	1.87218i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−18.0000	+	20.7846i	−0.807410	+	0.932317i
$498$		0			0
$499$			10.3923i			0.465223i	0.972570	+	0.232612i	$0.0747271\pi$
$499$			10.3923i			0.465223i	−0.972570	+	0.232612i	$0.925273\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	24.0000			1.07011			0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000			1.07011			0.535054	+	0.844818i	$0.320291\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	27.7128i		−	1.22835i	−0.789170	−	0.614174i	$-0.789489\pi$
$509$		−	27.7128i		−	1.22835i	0.789170	−	0.614174i	$-0.210511\pi$
$510$		0			0
$511$	−24.0000	+	27.7128i	−1.06170	+	1.22594i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$			6.92820i			0.303530i	0.988417	+	0.151765i	$0.0484957\pi$
$521$			6.92820i			0.303530i	−0.988417	+	0.151765i	$0.951504\pi$
$522$		0			0
$523$	20.0000			0.874539			0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000			0.874539			0.437269	+	0.899331i	$0.355946\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			27.7128i			1.20719i
$528$		0			0
$529$	11.0000			0.478261
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	24.0000	−	3.46410i	1.03375	−	0.149209i
$540$		0			0
$541$	2.00000			0.0859867			0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000			0.0859867			0.0429934	+	0.999075i	$0.486311\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$			24.2487i			1.03680i	0.855138	+	0.518400i	$0.173472\pi$
$547$			24.2487i			1.03680i	−0.855138	+	0.518400i	$0.826528\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−24.0000			−1.02243
$552$		0			0
$553$	18.0000	−	20.7846i	0.765438	−	0.883852i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−6.00000			−0.254228			−0.127114	−	0.991888i	$-0.540571\pi$
$557$	−6.00000			−0.254228			−0.127114	+	0.991888i	$0.540571\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−36.0000			−1.51722			−0.758610	−	0.651546i	$-0.774121\pi$
$563$	−36.0000			−1.51722			−0.758610	+	0.651546i	$0.774121\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	6.00000			0.251533			0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000			0.251533			0.125767	+	0.992060i	$0.459861\pi$
$570$		0			0
$571$		−	17.3205i		−	0.724841i	−0.932015	−	0.362420i	$-0.881950\pi$
$571$		−	17.3205i		−	0.724841i	0.932015	−	0.362420i	$-0.118050\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			17.3205i			0.722315i
$576$		0			0
$577$			13.8564i			0.576850i	0.957503	+	0.288425i	$0.0931316\pi$
$577$			13.8564i			0.576850i	−0.957503	+	0.288425i	$0.906868\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−24.0000	−	20.7846i	−0.995688	−	0.862291i
$582$		0			0
$583$			20.7846i			0.860811i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−12.0000			−0.495293			−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000			−0.495293			−0.247647	+	0.968850i	$0.579657\pi$
$588$		0			0
$589$	16.0000			0.659269
$590$		0			0
$591$		0			0
$592$		0			0
$593$			6.92820i			0.284507i	0.989830	+	0.142254i	$0.0454349\pi$
$593$			6.92820i			0.284507i	−0.989830	+	0.142254i	$0.954565\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	10.3923i		−	0.424618i	−0.977203	−	0.212309i	$-0.931902\pi$
$599$		−	10.3923i		−	0.424618i	0.977203	−	0.212309i	$-0.0680983\pi$
$600$		0			0
$601$			27.7128i			1.13043i	0.824944	+	0.565215i	$0.191207\pi$
$601$			27.7128i			1.13043i	−0.824944	+	0.565215i	$0.808793\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−4.00000			−0.162355			−0.0811775	−	0.996700i	$-0.525868\pi$
$607$	−4.00000			−0.162355			−0.0811775	+	0.996700i	$0.525868\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	10.0000			0.403896			0.201948	−	0.979396i	$-0.435273\pi$
$613$	10.0000			0.403896			0.201948	+	0.979396i	$0.435273\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	6.00000			0.241551			0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000			0.241551			0.120775	+	0.992680i	$0.461462\pi$
$618$		0			0
$619$	−4.00000			−0.160774			−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000			−0.160774			−0.0803868	+	0.996764i	$0.525616\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−12.0000	+	13.8564i	−0.480770	+	0.555145i
$624$		0			0
$625$	25.0000			1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$			13.8564i			0.552491i
$630$		0			0
$631$		−	3.46410i		−	0.137904i	−0.997620	−	0.0689519i	$-0.978035\pi$
$631$		−	3.46410i		−	0.137904i	0.997620	−	0.0689519i	$-0.0219655\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	6.00000			0.236986			0.118493	−	0.992955i	$-0.462194\pi$
$641$	6.00000			0.236986			0.118493	+	0.992955i	$0.462194\pi$
$642$		0			0
$643$	−44.0000			−1.73519			−0.867595	−	0.497271i	$-0.834335\pi$
$643$	−44.0000			−1.73519			−0.867595	+	0.497271i	$0.834335\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$		0			0
$649$		−	41.5692i		−	1.63173i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−6.00000			−0.234798			−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000			−0.234798			−0.117399	+	0.993085i	$0.537456\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$			24.2487i			0.944596i	0.881439	+	0.472298i	$0.156575\pi$
$659$			24.2487i			0.944596i	−0.881439	+	0.472298i	$0.843425\pi$
$660$		0			0
$661$		−	27.7128i		−	1.07790i	−0.842337	−	0.538952i	$-0.818821\pi$
$661$		−	27.7128i		−	1.07790i	0.842337	−	0.538952i	$-0.181179\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		−	20.7846i		−	0.804783i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−48.0000			−1.85302
$672$		0			0
$673$	34.0000			1.31060			0.655302	−	0.755367i	$-0.272541\pi$
$673$	34.0000			1.31060			0.655302	+	0.755367i	$0.272541\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			13.8564i			0.532545i	0.963898	+	0.266272i	$0.0857921\pi$
$677$			13.8564i			0.532545i	−0.963898	+	0.266272i	$0.914208\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$			24.2487i			0.927851i	0.885874	+	0.463926i	$0.153559\pi$
$683$			24.2487i			0.927851i	−0.885874	+	0.463926i	$0.846441\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−4.00000			−0.152167			−0.0760836	−	0.997101i	$-0.524242\pi$
$691$	−4.00000			−0.152167			−0.0760836	+	0.997101i	$0.524242\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−48.0000			−1.81813
$698$		0			0
$699$		0			0
$700$		0			0
$701$	18.0000			0.679851			0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000			0.679851			0.339925	+	0.940452i	$0.389598\pi$
$702$		0			0
$703$	8.00000			0.301726
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−24.0000	+	27.7128i	−0.902613	+	1.04225i
$708$		0			0
$709$	34.0000			1.27690			0.638448	−	0.769665i	$-0.279577\pi$
$709$	34.0000			1.27690			0.638448	+	0.769665i	$0.279577\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			13.8564i			0.518927i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−48.0000			−1.79010			−0.895049	−	0.445968i	$-0.852860\pi$
$719$	−48.0000			−1.79010			−0.895049	+	0.445968i	$0.852860\pi$
$720$		0			0
$721$	−8.00000	−	6.92820i	−0.297936	−	0.258020i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−30.0000			−1.11417
$726$		0			0
$727$	−20.0000			−0.741759			−0.370879	−	0.928681i	$-0.620944\pi$
$727$	−20.0000			−0.741759			−0.370879	+	0.928681i	$0.620944\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	24.0000			0.887672
$732$		0			0
$733$		−	27.7128i		−	1.02360i	−0.859106	−	0.511798i	$-0.828980\pi$
$733$		−	27.7128i		−	1.02360i	0.859106	−	0.511798i	$-0.171020\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−12.0000			−0.442026
$738$		0			0
$739$		−	3.46410i		−	0.127429i	−0.997968	−	0.0637145i	$-0.979705\pi$
$739$		−	3.46410i		−	0.127429i	0.997968	−	0.0637145i	$-0.0202947\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			3.46410i			0.127086i	0.997979	+	0.0635428i	$0.0202399\pi$
$743$			3.46410i			0.127086i	−0.997979	+	0.0635428i	$0.979760\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−30.0000	+	34.6410i	−1.09618	+	1.26576i
$750$		0			0
$751$		−	17.3205i		−	0.632034i	−0.948753	−	0.316017i	$-0.897654\pi$
$751$		−	17.3205i		−	0.632034i	0.948753	−	0.316017i	$-0.102346\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	34.0000			1.23575			0.617876	−	0.786276i	$-0.287994\pi$
$757$	34.0000			1.23575			0.617876	+	0.786276i	$0.287994\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	6.92820i		−	0.251147i	−0.992084	−	0.125574i	$-0.959923\pi$
$761$		−	6.92820i		−	0.251147i	0.992084	−	0.125574i	$-0.0400771\pi$
$762$		0			0
$763$	−20.0000	−	17.3205i	−0.724049	−	0.627044i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			27.7128i			0.996761i	0.866959	+	0.498380i	$0.166072\pi$
$773$			27.7128i			0.996761i	−0.866959	+	0.498380i	$0.833928\pi$
$774$		0			0
$775$	20.0000			0.718421
$776$		0			0
$777$		0			0
$778$		0			0
$779$			27.7128i			0.992915i
$780$		0			0
$781$	−36.0000			−1.28818
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−4.00000			−0.142585			−0.0712923	−	0.997455i	$-0.522712\pi$
$787$	−4.00000			−0.142585			−0.0712923	+	0.997455i	$0.522712\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−12.0000	−	10.3923i	−0.426671	−	0.369508i
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$			27.7128i			0.981638i	0.871262	+	0.490819i	$0.163302\pi$
$797$			27.7128i			0.981638i	−0.871262	+	0.490819i	$0.836698\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−48.0000			−1.69388
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−42.0000			−1.47664			−0.738321	−	0.674450i	$-0.764381\pi$
$809$	−42.0000			−1.47664			−0.738321	+	0.674450i	$0.764381\pi$
$810$		0			0
$811$	20.0000			0.702295			0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000			0.702295			0.351147	+	0.936320i	$0.385792\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		−	13.8564i		−	0.484774i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−6.00000			−0.209401			−0.104701	−	0.994504i	$-0.533388\pi$
$821$	−6.00000			−0.209401			−0.104701	+	0.994504i	$0.533388\pi$
$822$		0			0
$823$			51.9615i			1.81126i	0.424064	+	0.905632i	$0.360603\pi$
$823$			51.9615i			1.81126i	−0.424064	+	0.905632i	$0.639397\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	17.3205i		−	0.602293i	−0.953578	−	0.301147i	$-0.902631\pi$
$827$		−	17.3205i		−	0.602293i	0.953578	−	0.301147i	$-0.0973693\pi$
$828$		0			0
$829$			41.5692i			1.44376i	0.692019	+	0.721879i	$0.256721\pi$
$829$			41.5692i			1.44376i	−0.692019	+	0.721879i	$0.743279\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−48.0000	+	6.92820i	−1.66310	+	0.240048i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	48.0000			1.65714			0.828572	−	0.559883i	$-0.189154\pi$
$839$	48.0000			1.65714			0.828572	+	0.559883i	$0.189154\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	2.00000	+	1.73205i	0.0687208	+	0.0595140i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			6.92820i			0.237496i
$852$		0			0
$853$			13.8564i			0.474434i	0.971457	+	0.237217i	$0.0762353\pi$
$853$			13.8564i			0.474434i	−0.971457	+	0.237217i	$0.923765\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	34.6410i		−	1.18331i	−0.806190	−	0.591657i	$-0.798474\pi$
$857$		−	34.6410i		−	1.18331i	0.806190	−	0.591657i	$-0.201526\pi$
$858$		0			0
$859$	−20.0000			−0.682391			−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000			−0.682391			−0.341196	+	0.939992i	$0.610832\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	10.3923i		−	0.353758i	−0.984233	−	0.176879i	$-0.943400\pi$
$863$		−	10.3923i		−	0.353758i	0.984233	−	0.176879i	$-0.0566002\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	36.0000			1.22122
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−14.0000			−0.472746			−0.236373	−	0.971662i	$-0.575959\pi$
$877$	−14.0000			−0.472746			−0.236373	+	0.971662i	$0.575959\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			20.7846i			0.700251i	0.936703	+	0.350126i	$0.113861\pi$
$881$			20.7846i			0.700251i	−0.936703	+	0.350126i	$0.886139\pi$
$882$		0			0
$883$			10.3923i			0.349729i	0.984593	+	0.174864i	$0.0559487\pi$
$883$			10.3923i			0.349729i	−0.984593	+	0.174864i	$0.944051\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	24.0000			0.805841			0.402921	−	0.915235i	$-0.367995\pi$
$887$	24.0000			0.805841			0.402921	+	0.915235i	$0.367995\pi$
$888$		0			0
$889$	−30.0000	+	34.6410i	−1.00617	+	1.16182i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−24.0000			−0.800445
$900$		0			0
$901$		−	41.5692i		−	1.38487i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$			51.9615i			1.72535i	0.505755	+	0.862677i	$0.331214\pi$
$907$			51.9615i			1.72535i	−0.505755	+	0.862677i	$0.668786\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			31.1769i			1.03294i	0.856306	+	0.516469i	$0.172754\pi$
$911$			31.1769i			1.03294i	−0.856306	+	0.516469i	$0.827246\pi$
$912$		0			0
$913$		−	41.5692i		−	1.37574i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	24.0000	+	20.7846i	0.792550	+	0.686368i
$918$		0			0
$919$		−	31.1769i		−	1.02843i	−0.857661	−	0.514216i	$-0.828083\pi$
$919$		−	31.1769i		−	1.02843i	0.857661	−	0.514216i	$-0.171917\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	10.0000			0.328798
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	6.92820i		−	0.227307i	−0.993520	−	0.113653i	$-0.963745\pi$
$929$		−	6.92820i		−	0.227307i	0.993520	−	0.113653i	$-0.0362554\pi$
$930$		0			0
$931$	4.00000	+	27.7128i	0.131095	+	0.908251i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	13.8564i		−	0.452669i	−0.974050	−	0.226335i	$-0.927326\pi$
$937$		−	13.8564i		−	0.452669i	0.974050	−	0.226335i	$-0.0726743\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	27.7128i		−	0.903412i	−0.892167	−	0.451706i	$-0.850816\pi$
$941$		−	27.7128i		−	0.903412i	0.892167	−	0.451706i	$-0.149184\pi$
$942$		0			0
$943$	−24.0000			−0.781548
$944$		0			0
$945$		0			0
$946$		0			0
$947$			51.9615i			1.68852i	0.535932	+	0.844261i	$0.319960\pi$
$947$			51.9615i			1.68852i	−0.535932	+	0.844261i	$0.680040\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	30.0000			0.971795			0.485898	−	0.874016i	$-0.338493\pi$
$953$	30.0000			0.971795			0.485898	+	0.874016i	$0.338493\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	36.0000	+	31.1769i	1.16250	+	1.00676i
$960$		0			0
$961$	−15.0000			−0.483871
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$			24.2487i			0.779786i	0.920860	+	0.389893i	$0.127488\pi$
$967$			24.2487i			0.779786i	−0.920860	+	0.389893i	$0.872512\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−36.0000			−1.15529			−0.577647	−	0.816286i	$-0.696029\pi$
$971$	−36.0000			−1.15529			−0.577647	+	0.816286i	$0.696029\pi$
$972$		0			0
$973$	8.00000	+	6.92820i	0.256468	+	0.222108i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−18.0000			−0.575871			−0.287936	−	0.957650i	$-0.592969\pi$
$977$	−18.0000			−0.575871			−0.287936	+	0.957650i	$0.592969\pi$
$978$		0			0
$979$	−24.0000			−0.767043
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−24.0000			−0.765481			−0.382741	−	0.923856i	$-0.625020\pi$
$983$	−24.0000			−0.765481			−0.382741	+	0.923856i	$0.625020\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	12.0000			0.381578
$990$		0			0
$991$		−	17.3205i		−	0.550204i	−0.961415	−	0.275102i	$-0.911288\pi$
$991$		−	17.3205i		−	0.550204i	0.961415	−	0.275102i	$-0.0887116\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$			41.5692i			1.31651i	0.752795	+	0.658255i	$0.228705\pi$
$997$			41.5692i			1.31651i	−0.752795	+	0.658255i	$0.771295\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4032.2.b.c.3583.1		2
3.2	odd	2		1344.2.b.c.895.1		2
4.3	odd	2		4032.2.b.g.3583.2		2
7.6	odd	2		4032.2.b.g.3583.1		2
8.3	odd	2		1008.2.b.h.559.2		2
8.5	even	2		1008.2.b.a.559.1		2
12.11	even	2		1344.2.b.b.895.2		2
21.20	even	2		1344.2.b.b.895.1		2
24.5	odd	2		336.2.b.a.223.1	✓	2
24.11	even	2		336.2.b.d.223.2	yes	2
28.27	even	2	inner	4032.2.b.c.3583.2		2
56.13	odd	2		1008.2.b.h.559.1		2
56.27	even	2		1008.2.b.a.559.2		2
84.83	odd	2		1344.2.b.c.895.2		2
168.5	even	6		2352.2.bl.d.31.1		2
168.11	even	6		2352.2.bl.d.607.1		2
168.53	odd	6		2352.2.bl.i.607.1		2
168.59	odd	6		2352.2.bl.j.607.1		2
168.83	odd	2		336.2.b.a.223.2	yes	2
168.101	even	6		2352.2.bl.c.607.1		2
168.107	even	6		2352.2.bl.c.31.1		2
168.125	even	2		336.2.b.d.223.1	yes	2
168.131	odd	6		2352.2.bl.i.31.1		2
168.149	odd	6		2352.2.bl.j.31.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.2.b.a.223.1	✓	2	24.5	odd	2
336.2.b.a.223.2	yes	2	168.83	odd	2
336.2.b.d.223.1	yes	2	168.125	even	2
336.2.b.d.223.2	yes	2	24.11	even	2
1008.2.b.a.559.1		2	8.5	even	2
1008.2.b.a.559.2		2	56.27	even	2
1008.2.b.h.559.1		2	56.13	odd	2
1008.2.b.h.559.2		2	8.3	odd	2
1344.2.b.b.895.1		2	21.20	even	2
1344.2.b.b.895.2		2	12.11	even	2
1344.2.b.c.895.1		2	3.2	odd	2
1344.2.b.c.895.2		2	84.83	odd	2
2352.2.bl.c.31.1		2	168.107	even	6
2352.2.bl.c.607.1		2	168.101	even	6
2352.2.bl.d.31.1		2	168.5	even	6
2352.2.bl.d.607.1		2	168.11	even	6
2352.2.bl.i.31.1		2	168.131	odd	6
2352.2.bl.i.607.1		2	168.53	odd	6
2352.2.bl.j.31.1		2	168.149	odd	6
2352.2.bl.j.607.1		2	168.59	odd	6
4032.2.b.c.3583.1		2	1.1	even	1	trivial
4032.2.b.c.3583.2		2	28.27	even	2	inner
4032.2.b.g.3583.1		2	7.6	odd	2
4032.2.b.g.3583.2		2	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4032.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.00000 - 1.73205i) q^{7} +O(q^{10})\)	\(q+(-2.00000 - 1.73205i) q^{7} -3.46410i q^{11} +6.92820i q^{17} +4.00000 q^{19} +3.46410i q^{23} +5.00000 q^{25} -6.00000 q^{29} +4.00000 q^{31} +2.00000 q^{37} +6.92820i q^{41} -3.46410i q^{43} +(1.00000 + 6.92820i) q^{49} -6.00000 q^{53} +12.0000 q^{59} -13.8564i q^{61} -3.46410i q^{67} -10.3923i q^{71} -13.8564i q^{73} +(-6.00000 + 6.92820i) q^{77} +10.3923i q^{79} +12.0000 q^{83} -6.92820i q^{89} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{7}+O(q^{10}) \) 2 * q - 4 * q^7	\( 2 q - 4 q^{7} + 8 q^{19} + 10 q^{25} - 12 q^{29} + 8 q^{31} + 4 q^{37} + 2 q^{49} - 12 q^{53} + 24 q^{59} - 12 q^{77} + 24 q^{83}+O(q^{100}) \) 2 * q - 4 * q^7 + 8 * q^19 + 10 * q^25 - 12 * q^29 + 8 * q^31 + 4 * q^37 + 2 * q^49 - 12 * q^53 + 24 * q^59 - 12 * q^77 + 24 * q^83

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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